Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

Abstract

A plasma-sheath transition analysis requires a reliable mathematical expression for the plasma potential profile UðxÞ near the sheath edge x s in the limit e k D =' ¼ 0 (where k D is the Debye length and ' is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation (e ¼ 0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys. D: Appl. Phys. 24, 493 (1991)] only predicted a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann's formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since "the accuracy of the classical solutions is not sufficient to analyze the sheath vicinity" [Riemann, in ]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemann's general formula regarding the potential profile at the sheath edge in the cases of regular very warm ion sources. Fortunately, now we are able to achieve a very high accuracy of results [see, e.g., Kos et al., Phys. Plasmas 16, 093503 (2009)]. We perform this task by using both the analytic and the numerical method with explicit Maxwellian and "water-bag" ion source velocity distributions. We find the potential profile near the plasma-sheath edge in the whole range of ion source temperatures of general interest to plasma physics, from zero to "practical infinity." While within limits of "very low" and "relatively high" ion source temperatures, the potential is proportional to the space coordinate powered by rational numbers a ¼ 1=2 and a ¼ 2=3, with medium ion source temperatures. We found a between these values being a non-rational number strongly dependent on the ion source temperature. The range of the non-rational power-law turns out to be a very narrow one, at the expense of the extension of a ¼ 2=3 region towards unexpectedly low ion source temperatures.

A unified analysis of plasma-sheath transition in the Tonks–Langmuir model with warm ion source

“…U s (we remind that U 0 and U s 0) and are in accordance with the results of Ref. 10. Quite formally, the relations (41) and (42) can be extended for the region where U U s , assuming…”

“…To present time, several papers [8][9][10][11][12][13] have been published, where the thermal motion of the neutral gas is taken into account. But, in all these papers, the characteristics of only quasineutral presheath are investigated (In citations above, to our knowledge, the complete number of papers dedicated to the problem is given).…”

Potential Formation in Front of an Electrode Close to the Plasma Potential Studied by PIC Simulation

“…The obtained results are in very good agreement with the model of Jelić [2] made for cold ion source in collision free plasma model. This gives a hope that the code could be used also for the studies of sheath formation in the presence of warm ion sources [19][20][21] and/or magnetic field [22][23][24], where at the moment we are still lacking reliable theoretical models. There are of course still some open questions related to the consistency of our simulations.…”

“…It has been shown that the results of this code are in very good agreement with the analytical solutions [11] of the Tonks-Langmuir model for the ions born at rest. The code could very probably also be used for the study of sheath formation in more fusion relevant conditions with warm ions [19][20][21] and magnetic field [22][23][24] present. As we have shown above, a small potential difference between the electrode and the plasma can be achieved if a certain reflectivity r for the electrons is introduced for the selected -usually left -electrode.…”

Numerical analysis of ion temperature effects to the plasma wall transition using a one-dimensional two-fluid model. I. Finite Debye to ionization length ratio

“…7 and 8 and many relevant references can be found in Ref. 8. It is, therefore, not surprising that the problem of the ion temperature effects to the plasma-sheath formation is still mainly studied with one dimensional steady-state fluid models.…”